Draft:Chapman-Kolmogorov differential form

• Comment: In accordance with Wikipedia's Conflict of interest policy, I disclose that I have a conflict of interest regarding the subject of this article. Tommaso Talli (talk) 18:55, 6 November 2025 (UTC)

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The differential form of the Chapman–Kolmogorov equation describes the temporal evolution of the transition probability $P(x,t\mid x_{0},t_{0})$ of a Markov process, making explicit the relationship between the drift, diffusion, and transition rate terms in a stochastic model.

General form

The Chapman-Kolmogorov equation imposes strict limits on the general form that the transition probability $P(x,t|x_{0},t_{0})$ can take. In general, if $P(x,t|x_{0},t_{0})$ describes a Markov process, it must satisfy a well-defined partial differential equation.

${\begin{aligned}{\frac {\partial }{\partial t}}P(x,t|x_{0},t_{0})=&\underbrace {-\sum _{i}{\frac {\partial }{\partial x_{i}}}[A_{i}(x,t)P(x,t|x_{0},t_{0})]} _{\text{Drift term (continuous)}}\\[4pt]&\underbrace {+{\frac {1}{2}}\sum _{i,j}{\frac {\partial ^{2}}{\partial x_{i}\partial x_{j}}}[B_{ij}(x,t)P(x,t|x_{0},t_{0})]} _{\text{Diffusion term (continuous)}}\\[4pt]&\underbrace {+\int \mathrm {d} x'\,[W(x|x',t)P(x',t|x_{0},t_{0})-W(x'|x,t)P(x,t|x_{0},t_{0})]} _{\text{Jump term (disontinuous)}}\end{aligned}}$

where:

• $A_{i}$ are the drift coefficients,
• $B_{ij}$ is the diffusion matrix, and
• $W(x|x',t)$ is the jump rate from state $x'\mapsto x$.

The motion described by this partial differential equation can be interpreted as composed of a continuous part (drift and diffusion) and a discontinuous part (jump term).

Derivation from Chapman-Kolmogorov equation

Mathematical Definition of a Continuous Markov Process

For a Markov process the sample paths are continuous functions of t, if for any ε > 0 we have

$\lim _{\Delta t\to 0}{\frac {1}{\Delta t}}\int _{|x-z|>\varepsilon }dx\,p(x,t+\Delta t\mid z,t)=0$

uniformly in z, t and Δt.

This equation means that the probability for the final position x to be finitely different from z goes to zero faster than Δt, as Δt goes to zero. Because of the form of this continuity condition, one is led to consider a method of dividing the differentiability conditions into parts: one corresponding to continuous motion of a representative point and the other to discontinuous motion.

We require the following conditions for all ε > 0:

i) $\lim _{\Delta t\to 0}p(x,t+\Delta t\mid z,t)/\Delta t=W(x\mid z,t)$ uniformly in x, z, and t for |x − z| ≥ ε;

ii) $\lim _{\Delta t\to 0}{\frac {1}{\Delta t}}\int _{|x-z|<\varepsilon }dx\,(x_{i}-z_{i})\,p(x,t+\Delta t\mid z,t)=A_{i}(z,t)+O(\varepsilon )$

iii) $\lim _{\Delta t\to 0}{\frac {1}{\Delta t}}\int _{|x-z|<\varepsilon }dx\,(x_{i}-z_{i})(x_{j}-z_{j})\,p(x,t+\Delta t\mid z,t)=B_{ij}(z,t)+O(\varepsilon )$

The last two limits are uniform in z, ε, and t. Notice that all higher-order coefficients of the form in ii) and iii) must vanish.

Derivation of the Differential Chapman–Kolmogorov Equation

We consider the time evolution of the expectation of a function f(z) which is twice continuously differentiable.

Thus:

${\begin{aligned}\partial _{t}\int dx\,f(x)p(x,t\mid y,t')&=\lim _{\Delta t\to 0}{\frac {1}{\Delta t}}\left\{\int dx\,[f(x)p(x,t+\Delta t\mid y,t')-p(x,t\mid y,t')]\right\}\\&=\lim _{\Delta t\to 0}{\frac {1}{\Delta t}}{\Big \{}\int dx\,\int dz\,f(x)\,p(x,t+\Delta t\mid z,t)\,p(z,t\mid y,t')\\&-\int dz\,f(z)\,p(z,t\mid y,t'){\Big \}}\end{aligned}}$

where we have used the Chapman–Kolmogorov equation in the positive term of first step to produce the corresponding term in second step.

We now divide the integral over x into two regions: |x − z| ≥ ε and |x − z| < ε. When |x − z| < ε, since f(z) is, by assumption, twice continuously differentiable, we may write

$f(x)=f(z)+\sum _{i}{\frac {\partial f(z)}{\partial z_{i}}}(x_{i}-z_{i})+\sum _{i,j}{\frac {1}{2}}{\frac {\partial ^{2}f(z)}{\partial z_{i}\partial z_{j}}}(x_{i}-z_{i})(x_{j}-z_{j})+|x-z|^{2}R(x,z),$

where we have (by the twice continuous differentiability)

$|R(x,z)|\to 0\quad {\text{as}}\quad |x-z|\to 0.$

Now substitute $f(x)$ on the main formula:

${\begin{aligned}\partial _{t}\int dx\,f(x)p(x,t\mid y,t')=&\lim _{\Delta t\to 0}{\frac {1}{\Delta t}}{\Biggl \{}\iint _{|x-z|<\varepsilon }dx\,dz{\Bigl [}\sum _{i}(x_{i}-z_{i}){\frac {\partial f}{\partial z_{i}}}+\sum _{i,j}{\frac {1}{2}}(x_{i}-z_{i})(x_{j}-z_{j}){\frac {\partial ^{2}f}{\partial z_{i}\partial z_{j}}}{\Bigr ]}\\&\times p(x,t+\Delta t\mid z,t)p(z,t\mid y,t')\\&+\iint _{|x-z|<\varepsilon }dx\,dz\,|x-z|^{2}R(x,z)p(x,t+\Delta t\mid z,t)p(z,t\mid y,t')\\&+\iint _{|x-z|\geq \varepsilon }dx\,dz\,f(x)p(x,t+\Delta t\mid z,t)p(z,t\mid y,t')\\&+\iint _{|x-z|<\varepsilon }dx\,dz\,f(z)p(x,t+\Delta t\mid z,t)p(z,t\mid y,t')\\&-\iint dx\,dz\,f(z)p(x,t+\Delta t\mid z,t)p(z,t\mid y,t'){\Biggr \}}\end{aligned}}$ 

Lines 1–2: By the assumed uniform convergence, we take the limit inside the integral to obtain (using conditions (ii) and (iii)):

$\int dz{\Biggl [}\sum _{i}A_{i}(z){\frac {\partial f}{\partial z_{i}}}+{\frac {1}{2}}\sum _{i,j}B_{ij}(z){\frac {\partial ^{2}f}{\partial z_{i}\partial z_{j}}}{\Biggr ]}p(z,t\mid y,t')+O(\varepsilon ).$ 

Line 3: This is a remainder term and vanishes as $\varepsilon \to 0$. For

${\frac {1}{\Delta t}}\int _{|x-z|<\varepsilon }dx\,|x-z|^{2}R(x,z)p(x,t+\Delta t\mid z,t)\leq {\Biggl [}{\frac {1}{\Delta t}}\int _{|x-z|<\varepsilon }dx\,|x-z|^{2}p(x,t+\Delta t\mid z,t){\Biggr ]}\max _{|x-z|<\varepsilon }|R(x,z)|.$

Lines 4–6: We can put these all together to obtain

$\iint _{|x-z|>\varepsilon }dx\,dz\,f(z){\bigl [}W(x\mid z,t)p(x,t\mid y,t')-W(x\mid z,t)p(z,t\mid y,t'){\bigr ]}$

The whole right-hand side of the main formula is independent of ε. Hence, taking the limit ε → 0, we find

${\begin{aligned}\partial _{t}\int dz\,f(z)p(z,t\mid y,t')=&\int dz{\Biggl [}\sum _{i}A_{i}(z,t){\frac {\partial f(z)}{\partial z_{i}}}+{\frac {1}{2}}\sum _{i,j}B_{ij}(z,t){\frac {\partial ^{2}f(z)}{\partial z_{i}\partial z_{j}}}{\Biggr ]}p(z,t\mid y,t')\\+&\int dz\,f(z){\Biggl \{}{\text{P.V.}}\int dx{\bigl [}W(x\mid z,t)p(x,t\mid y,t')-W(x\mid z,t)p(z,t\mid y,t'){\bigr ]}{\Biggr \}}\end{aligned}}$

Notice, however, that we use the definition of a principal value:

$\lim _{\varepsilon \to 0}\int _{|x-z|>\varepsilon }dx\,F(x,z)\;\equiv \;{\text{P.V.}}\int dx\,F(x,z)$

The final step now is to integrate by parts. We find

${\begin{aligned}\int dz\,f(z)\partial _{t}p(z,t\mid y,t')&=\int dz\,f(z){\Biggl \{}-\sum _{i}{\frac {\partial }{\partial z_{i}}}{\bigl [}A_{i}(z,t)p(z,t\mid y,t'){\bigr ]}\\&\qquad \qquad \qquad +\sum _{i,j}{\frac {1}{2}}{\frac {\partial ^{2}}{\partial z_{i}\partial z_{j}}}{\bigl [}B_{ij}(z,t)p(z,t\mid y,t'){\bigr ]}\\[6pt]&\qquad \qquad \qquad +\int dx{\bigl [}W(x\mid z,t)p(x,t\mid y,t')-W(x\mid z,t)p(z,t\mid y,t'){\bigr ]}{\Biggr \}}\\&+{\text{surface terms}}\end{aligned}}$ 

Suppose the process is confined to a region $R$ with surface $S$. Then clearly,

$p(x,t\mid z,t')=0\quad {\text{unless both }}x,z\in R.$ 

It is clear that by definition we have

$W(x\mid z,t)=0\quad {\text{unless both }}x,y\in R.$ 

But the conditions on $A_{i}(z,t)$ and $B_{ij}(z,t)$ can result in discontinuities in these functions as defined by ii) and iii), since the conditional probability $p(x,t+\Delta t\mid z,t)$ can vary discontinuously as $z$ crosses the boundary of $R$, reflecting the fact that no transitions are allowed from outside $R$ to inside $R$.

In integrating by parts, we are forced to differentiate both $A_{i}$, and $B_{ij}$, and by our reasoning above, one cannot assume that this is possible on the boundary of the region. Hence, let us choose f(z) to be arbitrary but nonvanishing only in an arbitrary region $R'$ entirely contained in $R$. We can then deduce that for all z in the interior of $R$, surface terms do not arise, since they necessarily vanish.

So we obtain Chapman-Kolmogorov differential form. $\qquad \square$

Famous equations from Chapman-Kolmogorov differential form

We can identify three processes taking place, which are known as jumps, drift and diffusion.

Wiener process

The Wiener process in a Markov process is a special case of Chapman-Kolmogorov differential form equation in which there is only the drift term

${\frac {\partial }{\partial t}}P(x,t)={\frac {D}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}P(x,t)$

Fokker-Planck

The Fokker-Planck equation in a Markov process is a special case of Chapman-Kolmogorov differential form equation in which there is both a drift coefficient $A=\mu (x,t)$ and a diffusion coefficient $B=D(x,t)$ but there are no jumps.

${\frac {\partial }{\partial t}}P(x,t)=-{\frac {\partial }{\partial x}}\left[\mu (x,t)P(x,t)\right]+{\frac {\partial ^{2}}{\partial x^{2}}}\left[D(x,t)P(x,t)\right]$

Master equation

The Master equation in a Markov process is a special case of Chapman-Kolmogorov differential form equation in which only jumps are present.

${\frac {\partial }{\partial t}}P(x,t)=\int dx'{\Big [}W(x|x',t)P(x',t)-W(x|x',t)P(x',t){\Big ]}$

Liouville's Equation

The Liouville's equation in a Markov process is a special case of Chapman-Kolmogorov differential form equation in which only drift are present. This makes it a completely deterministic equation.

${\frac {\partial }{\partial t}}P(x,t)=-{\frac {\partial }{\partial x}}\left[\mu (x,t)P(x,t)\right]$

Displaying the terms of the equation

• $A_{i}\equiv$ drift coefficients: This is the average rate of change of the variable $z_{i}$ and generates a net shift in the probability distribution $P(z,t)$, in practice $A_{i}$ generates an ordered probability flow.

• $B_{ij}\equiv$ diffusion coefficients: This is the covariance of the noise and describes the intensity and correlation of the random fluctuations of the system, in practice $B_{ij}$ generates noise.

• $W(x|x',t)\equiv$ transition rate: This is the term that indicates the tendency of the system to transition from $x\mapsto x'$, in practice $W$ generates the jumps.

The form that the derivative of the probability can have based on the terms: drift, diffusion and jump.

See also

• Chapman-Kolmogorov equation
• Fokker–Planck equation
• Wiener process
• Master equation
• Liouville's theorem (Hamiltonian)
• Markov process